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Lecture Notes LECTURE NOTES College` de France Paris, France, November 2015 Lars-Erik Persson 1 Preface This ”Lecture notes” is a basic material written as a basis for the lectures The Hardy inequality- prehistory, history and current status and The interplay between Convexity, Interpolation and Inequalities presented at my visit at College` de France in November 2015 on invitation by Pro- fessor Pierre-Louis Lions. I cordially thank Professor Pierre-Louis Lions and College` de France for this kind invitation. I also thank Professor Natasha Samko, Lulea˚ University of Technology, for some related late joint research and for helping me to finalize this material. I hope this material can serve not only as a basis of these lectures but also as a source of inspiration for further research in this area. In particular, a number of open questions are pointed out. The material is closely connected to the following books: [1] A. Kufner and L.E. Persson, Weighted Inequalities of Hardy Type, World Scientific Pub- lishing Co. Inc., River Edge, NJ, 2003. [2] A. Kufner, L. Maligranda and L.E. Persson, The Hardy Inequality. About its History and Some Related Results, Vydavatelsky Servis Publishing House, Pilsen, 2007. [3] L. Larsson, L. Maligranda, J. Pecˇaric and L.E. Persson, Multiplicative Inequalities of Carlson Type and Interpolation, World Scientific Publishing Co., New Jersey-London-Singapore- Beijing-Shanghai-Hong Kong-Chennai, 2006. [4] C. Niculescu and L.E. Persson, Convex Functions and their Applications- A Contempo- rary Approach. Canad. Math. Series Books in Mathematics, Springer. 2006. [5] V. Kokilashvili, A. Meskhi and L.E. Persson, Weighted Norm Inequalities for Integral transforms with Product Weights, Nova Scientific Publishers, Inc., New York, 2010. But also some newer results and ideas can be found in this Lecture Notes, in particular from the following manuscript: [6] L.E. Persson and N. Samko, Classical and New Inequalities via Convexity and Interpola- tion, book manuscript, in preparation. College` de France, November, 2015 Lars-Erik Persson 2 3 LECTURE II The interplay between Convexity, Interpolation and Inequalities Lars-Erik Persson Lulea˚ University of Technology, Sweden, and Narvik University College, Norway [email protected] College` de France Paris, France, November 2015 1 More information concerning this lecture can be found in the following: [1] L. Larsson, L. Maligranda, J. Pecˇaric and L.E. Persson, Multiplicative Inequalities of Carl- son Type and Interpolation, World Scientific Publishing Co., New Jersey-London-Singapore- Beijing-Shanghai-Hong Kong-Chennai, 2006. [2] C. Niculescu and L.E. Persson, Convex Functions and their Applications- A Contemporary Approach. Canad. Math. Series Books in Mathematics, Springer. 2006. [3] L.E. Persson and N. Samko, Classical and New Inequalities via Convexity and Interpolation, book manuscript, in preparation. However, this lecture contains also some newer information, which can not be found in these books and the references given there. 2 1 A. CONVEXITY =) INEQUALITIES Let I denote a finite or infinite interval on R+: We say that a function f is convex on I if, for 0 < λ < 1; and all x; y 2 I; f((1 − λ)x + λy) ≤ (1 − λ)f(x) + λf(y): If this inequality holds in the reversed direction, then we say that the function f is concave. Examples of convex functions are f(x) = ex; x 2 R; f(x) = xa; x ≥ 0; a ≥ 1 or a < 0; and (1 + xp)1=p; x ≥ 0; p > 1: The notion of convexity (concavity) can be defined in a similar way for functions of more variables or even on more general sets. Here we just mention the following two-dimensional one by (the Swedish-Hungarian Professor) Marcel Riesz, which was very crucial when he proved his Riesz convexity theorem, which was very important when interpolation theory was initiated via the famous Riesz-Thorin interpolation theorem, see e.g. the book [B1] by J.Bergh and J. Lofstr¨ om.¨ Example A1 Let a and b be complex numbers. Then the function ( ) ja + bj1/α + ja − bj1/α α f(α; β) = log max (jaj1/β + jbj1/β)β is convex on the triangle T : 0 ≤ α ≤ β ≤ 1: Remark Another inportant student of Riesz was (the Swedish Professor) Lars Hormander,¨ which has written one of the most important books concerning the notion of convexity and its applications, see [C2]. 3 1.1 Convex functions at a first glance J. L. W. V. Jensen claimed: ”It seems to me that the notion of convex function is just as funda- mental as positive function or increasing function. If I am not mistaken in this, the notion ought to find a place in elementary expositions of the theory of real functions”. The following useful estimates are more or less easy consequences of the convexity (concav- ity) of the function f(x) = xp : Example A2 Let a1; a2; :::; an be positive numbers. Then ! Xn Xn p Xn p ≤ ≤ p−1 p ≥ (a) ai ai n ai ; p 1; i=1 i=1 i=1 ! Xn Xn p Xn p−1 p ≤ ≤ p ≤ (b) n ai ai ai ; 0 < p 1: i=1 i=1 i=1 The next example is a consequence of the convexity of the function f(x) = ex: 2 R n f g 1 1 Example A3 (Young’s inequality) For any a; b > 0; p; q 0 ; p + q = 1; it yields that ap bq ab ≤ + ; if p > 1; (1.1) p q and ap bq ab ≥ + ; if p < 1; p =6 f0g: (1.2) p q ”Proof of (1.1)”: ln ab 1 ln ap+ 1 ln bq 1 ln ap 1 ln bq 1 p 1 q ab = e = e p q ≤ e + e = a + b : p q p q Example A4 (Two fundamental inequalities” In the book [A3] by E.F. Beckenbach and R. Bellman it is claimed that the following inequalities are ”fundamental relations”: If x > 0 and α 2 R; then { xα − αx + α − 1 ≥ 0 for α > 1 and α < 0 (1.3) xα − αx + α − 1 ≤ 0 for 0 < α < 1: Remark In particular, they show later in the book that several well-known inequalities follow directly from (1.3) e.g. the AG-inequality, Holder’s¨ inequality, Minkowski’s inequality, etc. In the first lecture it was also pointed out that also Bennett’s inequalities (even in more precise form) of importance in interpolation theory follows in (1.3). In [A3] it was given two different proofs of (1.3) but indeed (1.3) follows directly from the fact that the function f(x) = xα is convex for α > 1 and α < 0 and concave for 0 < α < 1: In fact, if f(x) = xα; then the equation for the tangent at x = 1 is equal to l(x) = α(x − 1) + 1 and (1.3) follows directly. 4 ap 1 Note also that (1.1) follows directly from (1.3) applied with x = bq and α = p (the case 0 < α < 1) and (1.2) follows from (1.3) in the same way by instead applying (1.3) in the cases α > 1 and α < 0: We finish this Section by noting that also another useful inequality follows from convexity via Example A1. 2 ≤ p Example A5 Let a; b C; 1 < p 2 and q = p−1 ; then (ja + bjq + ja − bjq)1=q ≤ 21=q (jajp + jbjp)1=p : (1.4) The inequality is sharp, i.e. 21=q can not be replaced by any smaller number. One proof of (1.4) is to consider the convex function f(α; β) defined in Example A1. By using the parallelogram law ja + bj2 + ja − bj2 = 2(jaj2 + jbj2) (1.5) and the triangle inequality max(ja + bj; ja − bj) ≤ jaj + jbj; (1.6) and making some straightforward calculations the proof follows. However, the (scale of) in- equalities in (1.4) may also be regarded as natural ”intermediate inequalities” for the ”endpoint inequalities” (1.5) and (1.6). This can be proved exactly via interpolation (see page 24). 1.2 Convexity and Jensen’s inequality We state Jensen’s inequality in the following fairly general form: Theorem A1 (Jensen’s inequality) Let µ be a positive measure on a σ−algebra @ in a set Ω so that µ(Ω) =1. If f is a real µ−integrable function, if −∞ ≤ a < f(x) < b ≤ 1 for all x 2 Ω and if Ψ is convex on (a; b); then (Z ) Z Ψ fdµ ≤ Ψ(f)dµ. (1.7) Ω Ω If Ψ is concave, then (1.7) holds in the reversed direction. P R n RemarkP If Ω = +; n = 2; 3; :::; µ = k=1 λkδk (δk is the unity mass at t = k), λk > 0 n and k=1 λk = 1; then Jensen’s inequality (1.7) coincides with discrete Jensen’s inequality with f(k) = ak: Conversely, if we put the mass 1 − λ at x and λ at y (x; y 2 I) and assume that (1.7) holds for positive function Ψ; then, we have that Ψ((1 − λ)x + λy) ≤ (1 − λ)Ψ(x) + λΨ(y); i.e. the function Ψ is convex. These considerations show in fact that Jensen’s inequality is more or less equivalent to the notion of convexity. 5 1.3 Holder¨ type inequalities 1 1 Example A6 (Holder’s¨ inequality) Let p > 1 and p + q = 1: Then k k ≤ k k k k f L1 f Lp g Lq ; i.e. ( ) ( ) Z Z 1=p Z 1=q jfgjdµ ≤ jfjpdµ jgjqdµ : (1.8) Ω Ω Ω For the case 0 < p < 1 (1.8) holds in the reverse direction. The standard proof of (1.8) is obtained by just applying Young’s inequality (1.1) with a = f(x); b = g(x) and integrating.
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